Pseudosymmetric random matrices: Semi-Poisson and sub-Wigner statistics

Real nonsymmetric matrices may have either real or complex conjugate eigenvalues. These matrices can be seen to be pseudosymmetric as eta M eta(-1) = M-t, where the metric. could be secular (a constant matrix) or depending upon the matrix elements of M. Here we construct ensembles of a large number N of pseudosymmetric n x n (n large) matrices using N [n(n + 1)/2 <= N <= n(2)] independent and identically distributed random numbers as their elements. Based on our numerical calculations, we conjecture that for these ensembles the nearest level spacing distributions [NLSDs, p(s)] are sub-Wigner as p(abc) (s) = ase(-bsc) (0 < c < 2) and the distributions of their eigenvalues fit well to D(is an element of) = A[tanh{(is an element of+ B)/C} -tanh{(is an element of -B)/C}] (exceptions also discussed). These sub-Wigner NLSDs are encountered in Anderson metal-insulator transition and topological transitions in a Josephson junction. Interestingly, p(s) for c = 1 is called semi-Poisson, and we show that it lies close to the form p(s) = 0.59sK(0)(0.45s(2)) derived for the case of 2 x 2 pseudosymmetric matrix where the eigenvalues are most aptly conditionally real, E-1,E-2 = a +/- root b(2) - c(2), which represent characteristic coalescing of eigenvalues in parity-time (PT) -symmetric systems.